910

Chapter 20.

Less-Numerical Algorithms

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ica).

20.5 Arithmetic Coding

We saw in the previous section that a perfect (entropy-bounded) coding scheme

would use

L

i

= − log

2

p

i

bits to encode character

i (in the range 1 ≤ i ≤ N

ch

),

if

p

i

is its probability of occurrence. Huffman coding gives a way of rounding the

L

i

’s to close integer values and constructing a code with those lengths. Arithmetic

coding

[1]

, which we now discuss, actually does manage to encode characters using

noninteger numbers of bits! It also provides a convenient way to output the result
not as a stream of bits, but as a stream of symbols in any desired radix. This latter
property is particularly useful if you want, e.g., to convert data from bytes (radix
256) to printable ASCII characters (radix 94), or to case-independent alphanumeric
sequences containing only A-Z and 0-9 (radix 36).

In arithmetic coding, an input message of any length is represented as a real

number

R in the range 0 ≤ R < 1. The longer the message, the more precision

required of

R. This is best illustrated by an example, so let us return to the fictitious

language, Vowellish, of the previous section. Recall that Vowellish has a 5 character
alphabet (A, E, I, O, U), with occurrence probabilities 0.12, 0.42, 0.09, 0.30, and
0.07, respectively. Figure 20.5.1 shows how a message beginning “IOU” is encoded:
The interval [0

, 1) is divided into segments corresponding to the 5 alphabetical

characters; the length of a segment is the corresponding

p

i

. We see that the first

message character, “I”, narrows the range of

R to 0.37 ≤ R < 0.46. This interval is

now subdivided into five subintervals, again with lengths proportional to the

p

i

’s. The

second message character, “O”, narrows the range of

R to 0.3763 ≤ R < 0.4033.

The “U” character further narrows the range to 0

.37630 ≤ R < 0.37819. Any value

of

R in this range can be sent as encoding “IOU”. In particular, the binary fraction

.011000001 is in this range, so “IOU” can be sent in 9 bits. (Huffman coding took
10 bits for this example, see

§20.4.)

Of course there is the problem of knowing when to stop decoding. The fraction

.011000001 represents not simply “IOU,” but “IOU. . . ,” where the ellipses represent
an infinite string of successor characters.

To resolve this ambiguity, arithmetic

coding generally assumes the existence of a special

N

ch

+ 1th character, EOM

(end of message), which occurs only once at the end of the input. Since EOM
has a low probability of occurrence, it gets allocated only a very tiny piece of
the number line.

In the above example, we gave

R as a binary fraction. We could just as well

have output it in any other radix, e.g., base 94 or base 36, whatever is convenient
for the anticipated storage or communication channel.

You might wonder how one deals with the seemingly incredible precision

required of

R for a long message. The answer is that R is never actually represented

all at once. At any give stage we have upper and lower bounds for

R represented

as a finite number of digits in the output radix. As digits of the upper and lower
bounds become identical, we can left-shift them away and bring in new digits at the
low-significance end. The routines below have a parameter NWK for the number of
working digits to keep around. This must be large enough to make the chance of
an accidental degeneracy vanishingly small. (The routines signal if a degeneracy
ever occurs.) Since the process of discarding old digits and bringing in new ones is
performed identically on encoding and decoding, everything stays synchronized.

20.5 Arithmetic Coding

911

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Copyright (C) 1988-1992 by Cambridge University Press.

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ica).

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

A

E

I

O

U

A

E

I

O

U

A

E

I

O

U

A

E

I

O

U

0.46

0.42

0.41

0.37

0.385

0.380

0.4033

0.3763

0.37819

0.37630

0.3780

0.3764

0.44

0.43

0.390

0.395

0.400

0.3766

0.3768

0.3772

0.3774

0.3778

0.3776

0.45

0.40

0.39

0.38

0.3770

Figure 20.5.1.

Arithmetic coding of the message “IOU...”

in the fictitious language Vowellish.

Successive characters give successively finer subdivisions of the initial interval between 0 and 1. The final
value can be output as the digits of a fraction in any desired radix. Note how the subinterval allocated
to a character is proportional to its probability of occurrence.

The routine arcmak constructs the cumulative frequency distribution table used

to partition the interval at each stage. In the principal routine arcode, when an
interval of size jdif is to be partitioned in the proportions of some n to some ntot,
say, then we must compute (n*jdif)/ntot. With integer arithmetic, the numerator
is likely to overflow; and, unfortunately, an expression like jdif/(ntot/n) is not
equivalent. In the implementation below, we resort to double precision floating
arithmetic for this calculation. Not only is this inefficient, but different roundoff
errors can (albeit very rarely) make different machines encode differently, though any
one type of machine will decode exactly what it encoded, since identical roundoff
errors occur in the two processes. For serious use, one needs to replace this floating
calculation with an integer computation in a double register (not available to the
C programmer).

The internally set variable minint, which is the minimum allowed number

of discrete steps between the upper and lower bounds, determines when new low-
significance digits are added. minint must be large enough to provide resolution of
all the input characters. That is, we must have

p

i

× minint > 1 for all i. A value

of 100

N

ch

, or 1

.1/ min p

i

, whichever is larger, is generally adequate. However, for

safety, the routine below takes minint to be as large as possible, with the product
minint*nradd just smaller than overflow. This results in some time inefficiency,
and in a few unnecessary characters being output at the end of a message. You can

912

Chapter 20.

Less-Numerical Algorithms

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

Copyright (C) 1988-1992 by Cambridge University Press.

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ica).

decrease minint if you want to live closer to the edge.

A final safety feature in arcmak is its refusal to believe zero values in the table

nfreq; a 0 is treated as if it were a 1. If this were not done, the occurrence in a
message of a single character whose nfreq entry is zero would result in scrambling
the entire rest of the message. If you want to live dangerously, with a very slightly
more efficient coding, you can delete the IMAX( ,1) operation.

#include "nrutil.h"
#include <limits.h>

ANSI header file containing integer ranges.

#define MC 512
#ifdef ULONG_MAX

Maximum value of unsigned long.

#define MAXINT (ULONG_MAX >> 1)
#else
#define MAXINT 2147483647
#endif
Here

MC

is the largest anticipated value of

nchh

;

MAXINT

is a large positive integer that does

not overflow.

typedef struct {

unsigned long *ilob,*iupb,*ncumfq,jdif,nc,minint,nch,ncum,nrad;

} arithcode;

void arcmak(unsigned long nfreq[], unsigned long nchh, unsigned long nradd,

arithcode *acode)

Given a table

nfreq[1..nchh]

of the frequency of occurrence of

nchh

symbols, and given

a desired output radix

nradd

, initialize the cumulative frequency table and other variables for

arithmetic compression in the structure

acode

.

{

unsigned long j;

if (nchh > MC) nrerror("input radix may not exceed MC in arcmak.");
if (nradd > 256) nrerror("output radix may not exceed 256 in arcmak.");

acode->minint=MAXINT/nradd;
acode->nch=nchh;
acode->nrad=nradd;
acode->ncumfq[1]=0;
for (j=2;j<=acode->nch+1;j++)

acode->ncumfq[j]=acode->ncumfq[j-1]+IMAX(nfreq[j-1],1);

acode->ncum=acode->ncumfq[acode->nch+2]=acode->ncumfq[acode->nch+1]+1;

}

The structure acode must be defined and allocated in your main program with

statements like this:

#include "nrutil.h"
#define MC 512

Maximum anticipated value of nchh in arcmak.

#define NWK 20

Keep this value the same as in arcode, b elow.

typedef struct {

unsigned long *ilob,*iupb,*ncumfq,jdif,nc,minint,nch,ncum,nrad;

} arithcode;

...
arithcode acode;
...
acode.ilob=(unsigned long *)lvector(1,NWK);

Allocate space within acode.

acode.iupb=(unsigned long *)lvector(1,NWK);
acode.ncumfq=(unsigned long *)lvector(1,MC+2);

20.5 Arithmetic Coding

913

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

Copyright (C) 1988-1992 by Cambridge University Press.

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isit website

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ica).

Individual characters in a message are coded or decoded by the routine arcode,

which in turn uses the utility arcsum.

#include <stdio.h>
#include <stdlib.h>
#define NWK 20
#define JTRY(j,k,m) ((long)((((double)(k))*((double)(j)))/((double)(m))))
This macro is used to calculate

(k*j)/m

without overflow. Program efficiency can be improved

by substituting an assembly language routine that does integer multiply to a double register.

typedef struct {

unsigned long *ilob,*iupb,*ncumfq,jdif,nc,minint,nch,ncum,nrad;

} arithcode;

void arcode(unsigned long *ich, unsigned char **codep, unsigned long *lcode,

unsigned long *lcd, int isign, arithcode *acode)

Compress (

isign

= 1) or decompress (

isign

= −1) the single character

ich

into or out of

the character array

*codep[1..lcode]

, starting with byte

*codep[lcd]

and (if necessary)

incrementing

lcd

so that, on return,

lcd

points to the first unused byte in

*codep

. Note

that the structure

acode

contains both information on the code, and also state information on

the particular output being written into the array

*codep

. An initializing call with

isign=0

is required before beginning any

*codep

array, whether for encoding or decoding. This is in

addition to the initializing call to

arcmak

that is required to initialize the code itself. A call

with

ich=nch

(as set in

arcmak

) has the reserved meaning “end of message.”

{

void arcsum(unsigned long iin[], unsigned long iout[], unsigned long ja,

int nwk, unsigned long nrad, unsigned long nc);

void nrerror(char error_text[]);
int j,k;
unsigned long ihi,ja,jh,jl,m;

if (!isign) {

Initialize enough digits of the upper and lower bounds.

acode->jdif=acode->nrad-1;
for (j=NWK;j>=1;j--) {

acode->iupb[j]=acode->nrad-1;
acode->ilob[j]=0;
acode->nc=j;
if (acode->jdif > acode->minint) return;

Initialization complete.

acode->jdif=(acode->jdif+1)*acode->nrad-1;

}
nrerror("NWK too small in arcode.");

} else {

if (isign > 0) {

If encoding, check for valid input character.

if (*ich > acode->nch) nrerror("bad ich in arcode.");

}
else {

If decoding, locate the character ich by bisection.

ja=(*codep)[*lcd]-acode->ilob[acode->nc];
for (j=acode->nc+1;j<=NWK;j++) {

ja *= acode->nrad;
ja += ((*codep)[*lcd+j-acode->nc]-acode->ilob[j]);

}
ihi=acode->nch+1;
*ich=0;
while (ihi-(*ich) > 1) {

m=(*ich+ihi)>>1;
if (ja >= JTRY(acode->jdif,acode->ncumfq[m+1],acode->ncum))

*ich=m;

else ihi=m;

}
if (*ich == acode->nch) return;

Detected end of message.

}
Following code is common for encoding and decoding. Convert character ich to a new
subrange [ilob,iupb).

914

Chapter 20.

Less-Numerical Algorithms

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

Copyright (C) 1988-1992 by Cambridge University Press.

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readable files (including this one) to any server

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isit website

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ica).

jh=JTRY(acode->jdif,acode->ncumfq[*ich+2],acode->ncum);
jl=JTRY(acode->jdif,acode->ncumfq[*ich+1],acode->ncum);
acode->jdif=jh-jl;
arcsum(acode->ilob,acode->iupb,jh,NWK,acode->nrad,acode->nc);
arcsum(acode->ilob,acode->ilob,jl,NWK,acode->nrad,acode->nc);
How many leading digits to output (if encoding) or skip over?
for (j=acode->nc;j<=NWK;j++) {

if (*ich != acode->nch && acode->iupb[j] != acode->ilob[j]) break;
if (*lcd > *lcode) {

fprintf(stderr,"Reached the end of the ’code’ array.\n");
fprintf(stderr,"Attempting to expand its size.\n");
*lcode += *lcode/2;
if ((*codep=(unsigned char *)realloc(*codep,

(unsigned)(*lcode*sizeof(unsigned char)))) == NULL) {
nrerror("Size expansion failed");

}

}
if (isign > 0) (*codep)[*lcd]=(unsigned char)acode->ilob[j];
++(*lcd);

}
if (j > NWK) return;

Ran out of message. Did someone forget to encode a

terminating ncd?

acode->nc=j;
for(j=0;acode->jdif<acode->minint;j++)

How many digits to shift?

acode->jdif *= acode->nrad;

if (acode->nc-j < 1) nrerror("NWK too small in arcode.");
if (j) {

Shift them.

for (k=acode->nc;k<=NWK;k++) {

acode->iupb[k-j]=acode->iupb[k];
acode->ilob[k-j]=acode->ilob[k];

}

}
acode->nc -= j;
for (k=NWK-j+1;k<=NWK;k++) acode->iupb[k]=acode->ilob[k]=0;

}
return;

Normal return.

}

void arcsum(unsigned long iin[], unsigned long iout[], unsigned long ja,

int nwk, unsigned long nrad, unsigned long nc)

Used by

arcode

. Add the integer

ja

to the radix

nrad

multiple-precision integer

iin[nc..nwk]

.

Return the result in

iout[nc..nwk]

.

{

int j,karry=0;
unsigned long jtmp;

for (j=nwk;j>nc;j--) {

jtmp=ja;
ja /= nrad;
iout[j]=iin[j]+(jtmp-ja*nrad)+karry;
if (iout[j] >= nrad) {

iout[j] -= nrad;
karry=1;

} else karry=0;

}
iout[nc]=iin[nc]+ja+karry;

}

If radix-changing, rather than compression, is your primary aim (for example

to convert an arbitrary file into printable characters) then you are of course free to
set all the components of nfreq equal, say, to 1.

20.6 Arithmetic at Arbitrary Precision

915

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

Copyright (C) 1988-1992 by Cambridge University Press.

Programs Copyright (C) 1988-1992 by Numerical Recipes Software.

Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin

g of machine-

readable files (including this one) to any server

computer, is strictly prohibited. To order Numerical Recipes books

or CDROMs, v

isit website

http://www.nr.com or call 1-800-872-7423 (North America only),

or send email to [e-mail] (outside North Amer

ica).

CITED REFERENCES AND FURTHER READING:

Bell, T.C., Cleary, J.G., and Witten, I.H. 1990, Text Compression (Englewood Cliffs, NJ: Prentice-

Hall).

Nelson, M. 1991, The Data Compression Book (Redwood City, CA: M&T Books).

Witten, I.H., Neal, R.M., and Cleary, J.G. 1987, Communications of the ACM, vol. 30, pp. 520–

540. [1]

20.6 Arithmetic at Arbitrary Precision

Let’s compute the number

π to a couple of thousand decimal places. In doing

so, we’ll learn some things about multiple precision arithmetic on computers and
meet quite an unusual application of the fast Fourier transform (FFT). We’ll also
develop a set of routines that you can use for other calculations at any desired level
of arithmetic precision.

To start with, we need an analytic algorithm for

π. Useful algorithms

are quadratically convergent, i.e., they double the number of significant digits at
each iteration. Quadratically convergent algorithms for

π are based on the AGM

(arithmetic geometric mean) method, which also finds application to the calculation
of elliptic integrals (cf.

§6.11) and in advanced implementations of the ADI method

for elliptic partial differential equations (

§19.5). Borwein and Borwein

[1]

treat this

subject, which is beyond our scope here. One of their algorithms for

π starts with

the initializations

X

0

=

√

2

π

0

= 2 +

√

2

Y

0

=

4

√

2

(20.6.1)

and then, for

i = 0, 1, . . . , repeats the iteration

X

i+1

=

1
2



X

i

+

1

√

X

i



π

i+1

= π

i

X

i+1

+ 1

Y

i

+ 1



Y

i+1

=

Y

i



X

i+1

+

1



X

i+1

Y

i

+ 1

(20.6.2)

The value

π emerges as the limit π

∞

.

Now, to the question of how to do arithmetic to arbitrary precision: In a

high-level language like C, a natural choice is to work in radix (base) 256, so that
character arrays can be directly interpreted as strings of digits. At the very end of
our calculation, we will want to convert our answer to radix 10, but that is essentially
a frill for the benefit of human ears, accustomed to the familiar chant, “three point